In this activity, students calculate three different benchmark percentages—50%, 10%, and 75%—given three different values that correspond to 100%. Calculating the same benchmark percentage for different quantities encourages students to notice regularity through repeated reasoning (MP8). The goal is for students to generalize the patterns in their calculations and determine how to find those percentages when the 100% value is .

Tape diagrams that represent 50% of different quantities are given for the first set of questions. The diagrams illustrate the same structure across the three situations, encouraging students to look for and make use of structure in answering all questions (MP7).

The purpose of the discussion is to highlight the benchmark fractions that correspond to 50%, 10%, and 75%.

Invite students to share their generalizations for finding 50%, 10%, and 75% of any number. Highlight the following points:

• 50% of a quantity is of that quantity. We can calculate it by dividing the quantity by 2 or multiplying the quantity by . If no students bring up the latter, ask what number they could multiply the quantity by to get the same answer. (Either or 0.5 is fine.)

• 10% of a quantity is of that quantity. We can calculate it by dividing the quantity by 10 or multiplying the quantity by .

  

• 75% of a quantity is of that quantity. We can calculate it by dividing the quantity by 4 to find 25% of the quantity and then multiplying the result by 3. Another way is to multiply the quantity by .

  

If 25% is not mentioned by students when discussing 75% of a quantity, ask how they might find 25% of any quantity. Highlight that it can be calculated by dividing the quantity by 4 or multiplying it by .

In this activity, students find the values that correspond to 100% when different benchmark percentages are known. They answer questions such as: “9 is 50% of what number?” Students are asked to calculate these values mentally and to explain their reasoning. To encourage students to continue making connections with fractions, a tape diagram is given to represent the quantities in the first question.

Reasoning repeatedly about the same number (9) as different percentages encourages students to look for and make use of structure (MP7). In explaining their reasoning, students practice communicating with precision (MP6).

While some students may mentally calculate the answers quickly and be able to explain their reasoning abstractly, others may find it helpful to use diagrams to make sense of the questions or to explain their thinking. Consider asking: “How might you use a tape diagram to show 9 is 25% of something or 9 is 10% of something?” Encourage them to look for a pattern in how they partition their diagrams and in the computations they perform to answer the series of questions.

Before inviting students to share their responses, use Critique, Correct, Clarify to give students an opportunity to improve a sample written response to the first question by correcting errors, clarifying meaning, and adding details.

• Display this first draft:

  “9 is 50% of 4.5 because ”

  Ask, “What parts of this response are unclear, incorrect, or incomplete?” As students respond, annotate the display with 2–3 ideas to indicate the parts of the writing that could use improvement.

• Give students 2 minutes to work with a partner to revise the first draft.
• Select 1–2 individuals or groups to read their revised draft aloud slowly enough to record for all to see. Scribe as each student shares, then invite the whole class to contribute additional language and edits to make the final draft even more clear and more convincing.

Briefly discuss the responses to the questions about 9 being 25% and 10% of another number. Consider drawing a tape diagram that can illustrate why the responses are and , respectively.

Then, focus the discussion about how students reasoned about the questions about 75% and 150%. If no students use tape diagrams in their reasoning, consider drawing one to illustrate the relationship between the numbers in each question. (See examples in Student Response.)

Emphasize that benchmark fractions such as , , , and can be used to make sense of and solve problems about finding the value for 100%.

In this activity, students solve percentage problems in the context of shopping. They consider how discounts described as percentages translate into reduced prices and the other way around. (In other words, they find and where of is .) In each problem, students need to first determine what value is associated with 100% and reason accordingly.

Students may choose to reason using double number line diagrams, tape diagrams, tables, or no particular representation. For instance, to find 10% of \$15, they may first find 50% by dividing \$15 by 2, and then divide the resulting \$7.50 by 5 to obtain 10% of \$15. Those who recognize 10% as a benchmark percentage and know that it is one-tenth of 100% may find or .

As students work, monitor for students who use different strategies so that they can share later.

Select 2–3 students who used different representations and strategies to share their reasoning. As students explain, illustrate and display those representations for all to see.

If no students used fractions in their reasoning, ask them to discuss this idea. Highlight that we can think of 10% of 15 in terms of of 15 and that we can think of 6 as of 24.

If no students created double number line diagrams or tables, demonstrate how one of these representations could be used. Ask students if the same table or double number line diagram could be used to solve both problems and discuss why or why not. Emphasize that two separate double number line diagrams or tables are necessary because the value for 100% is different in each case.